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Ferroelectrics

Equation of State for Ising Lattices

J. García, Manuel I. Marqués, Julio A. Gonzalo

Published Jun 1, 2005• 0 citations
Publisher

Abstract

Accurate Monte Carlo data from a set of isotherms near the critical point are analyzed using two RG based complementary representations, given respectively in terms of ℏ = h /|t |β δ and = t /h 1/β δ . Scaling plots for data on simple cubic Ising lattices are compared with plots of ML β ν vs. |t |L 1/ν for increasing L values and with high quality experimental data on CrBr 3 . Finite size effects and the equation of state are discussed.

DOI
10.1080/00150190590926166

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Accurate Monte Carlo data from a set of isotherms near the critical point are analyzed using two RG based complementary representations, given respectively in terms of ℏ = h /|t |β δ and = t /h 1/β δ . Scaling plots for data on simple cubic Ising lattices are compared with plots of ML β ν vs. |t |L 1/ν for increasing L values and with high quality experimental data on CrBr 3 . Finite size effects and the equation of state are discussed.
Accurate Monte Carlo data from a set of isotherms near the critical point are analyzed using two RG based complementary representations, given respectively in terms of hˉ=h/∣t∣βδ\bar h=h/\mid t \mid^{βδ}hˉ=h/∣t∣βδ and τˉ=t/h1/βδ\bar τ=t/h^{1/βδ}τˉ=t/h1/βδ. Scaling plots for data on simple cubic Ising lattices are compared with plots of MLβνML^{βν}MLβν vs. ∣t∣L1/ν\mid t \mid L^{1/ν}∣t∣L1/ν for increasing LLL values and with high quality experimental data on CrBr3CrBr_{3}CrBr3​. Finite size effects and the equation of state are discussed.
A careful Monte Carlo investigation of the phase transition very close to the critical point (T -> Tc, H -> 0) in relatively large d = 3, s = 1/2 Ising lattices did produce critical exponents beta = 0.3126(4) =~ 5/16, delta^{-1} = 0.1997(4) =~ 1/5 and gamma_{3D} = 1.253(4) =~ 5/4. Our results indicate that, within experimental error, they are given by simple fractions corresponding to the linear interpolations between the respective two-dimensional (Onsager) and four-dimensional (mean field) critical exponents. An analysis of our inverse susceptibility data chi^{-1}(T) vs. /T - Tc/ shows that these data lead to a value of gamma_{3D} compatible with gamma' = gamma and Tc = 4.51152(12), while gamma values obtained recently by high and low temperature series expansions and renormalization group methods are not.
Monte Carlo investigations of magnetization versus field, Mc(H), at the critical temperature provide direct accurate results on the critical exponent δ−1 for one, two, three and four-dimensional lattices: δ−1 1D =0, δ−1 2D =0.0666(2)≈1/15, δ−1 3D =0.1997(4)≈1/5, δ−1 4D =0.332(5)≈1/3. This type of Monte Carlo data on δ, which is not easily found in studies of Ising lattices in the current literature, as far as we know, defines extremely well the numerical value of this exponent within very stringent limits.
Using extensive Monte Carlo simulations that employ the Wolff cluster flipping and data analysis with histogram reweighting and quadruple precision arithmetic, we have investigated the critical behavior of the simple cubic Ising model with lattice sizes ranging from 16 3 to 1024 3 . By analyzing data with cross correlations between various thermodynamic quantities obtained from the same data pool, we obtained the critical inverse temperature K c = 0.221 654 626(5) and the critical exponent of the correlation length ν = 0.629 912(86) with precision that improves upon previous Monte Carlo estimates.
Using recently developed histogram techniques and an ultrafast multispin coding simulation algorithm, we have investigated the critical behavior of the d=3 simple-cubic Ising model. We have studied lattice sizes ranging from L=8 to 96 using between 3\ifmmode\times\else\texttimes\fi{}106{10}^{6}106 and 12\ifmmode\times\else\texttimes\fi{}106{10}^{6}106 Monte Carlo steps (complete lattice updates). By accurately measuring the finite-size behavior of several different thermodynamic quantities, we are able to determine the critical properties with a precision comparable to that obtained with Monte Carlo renormalization-group and sophisticated series-expansion techniques. The best estimate of the inverse critical temperature from our analysis is Kc{\mathit{K}}_{\mathit{c}}Kc​=0.221 659 5\ifmmode\pm\else\textpm\fi{}0.000 002 6. The advantages of the histogram technique are discussed, as are the potential problems that can arise at this level of resolution.